Going Down the Exponential Slide

A bead slides under the pull of gravity, \(g,\) down a frictionless wire segment in the shape of the curve \(y = e^{-x}\), where \(x\) is the horizontal direction and \(y\) is the vertical direction. The bead starts from rest at \((x,y) = (0,1)\).

The time it takes for the particle to travel between \(x=a\) and \(x=b\) can be expressed as

\[\large{t_{a,b} = \frac{1}{\sqrt{2g}} \int_a^b \sqrt{\frac{1 + P e^{Q x}}{1 + R e^{S x} }}\,dx},\]

where \(P,Q,R,\) and \(S\) are integers.

Determine \(P + Q + R + S.\)

Note: The constant \(e\) is Euler's number.


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